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                                                                    Exhibit 10.1

                              PURE RESOURCES, INC.
                                500 West Illinois
                              Midland, Texas 79701

                                 April 23, 2002

Union Oil Company of California
2141 Rosecrans Avenue, Suite 4000
El Segundo, California 90245

     Re: Pure Resources, Inc. - Business Opportunities Agreement

Ladies and Gentlemen:

     Reference is made to the Business Opportunities Agreement (the "Agreement")
dated as of December 13, 1999 among Union Oil Company of California ("Union
Oil"), Pure Resources, Inc. (formerly named Titan Resources Holdings, Inc.) (the
"Company"), TRH, Inc., and Titan Exploration, Inc. Capitalized terms used and
not otherwise defined herein shall have the meanings given to them in the
Agreement. The Agreement was entered into to address certain legal requirements
and concerns as well as for certain practical business purposes. Although this
letter does request a limited waiver of the Agreement, we wish to make it clear
that we understand that the Agreement will remain in effect after any grant of a
limited waiver and that we will continue to remain focused on the business we
conduct in the Designated Area and any other area that you may permit us to
operate in pursuant to this or any other limited waiver.

     Pursuant to Section 1 of the Agreement, the Company agreed that, except
with the consent of Union Oil (which it may withhold in its sole discretion),
neither the Company nor its subsidiaries will engage in any business other than
the E&P Business and none of them will pursue any business opportunity that
involves any direct or indirect ownership interest in any properties located
outside the Designated Area.

     On May 9, 2001, you granted the Company a limited waiver of the Agreement
in connection with certain transactions (the "May 9 Limited Waiver"). We hereby
confirm that Agreement remains in full force and effect and that we are
continuing to observe and comply with its terms allowing for the provisions of
the May 9 Limited Waiver.

     The Company has now been presented with exploration opportunities in the
Gulf of Mexico that are not located within the parameters of the Offshore
Extended Business as defined in the May 9 Waiver. In connection therewith, we
hereby request that Union Oil agree that the term "Offshore Extended Business"
be deemed to include the Company's participation in a seven prospect Gulf of

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April 23,2002
Page 2

Mexico shelf drilling program generated and operated by Gryphon Exploration, the
details of which were communicated to Union Oil on April 22, 2002.

     By signing below, Union Oil hereby acknowledges that the Company will not
breach or violate the Agreement by pursuing or engaging in either the Onshore
Extended Business or the Offshore Extended Business (as such terms are defined
in the May 9 Limited Waiver and as the term "Offshore Extended Business" is
modified by this letter). Notwithstanding the foregoing, the limited waiver that
Union Oil grants hereby with respect to the Offshore Extended Business shall
expire and terminate on March 31, 2003 if Union Oil notifies the Company in
writing at any time in the month of September, 2002, that this limited waiver
shall so terminate and that Union Oil will not extend the effective period of
such limited waiver beyond March 31, 2003. If Union Oil notifies the Company in
writing, during the month of September, 2002, that the limited waiver for
Offshore Extended Business shall be extended beyond March 31, 2003, such limited
waiver will be so extended for the period of time set forth in Union Oil's
notification. Union Oil's right to so notify the Company to either terminate or
extend the limited waiver for Offshore Extended Business may be exercised in its
sole discretion.

     We hereby acknowledge that if you agree to the extension of "Offshore
Extended Business" as set forth above, that such extension in no way shall serve
to waive any provision of the Agreement that may be applicable to future
transactions or investments by the Company that are not included within the
Onshore Extended Business or the Offshore Extended Business (as such terms are
defined after execution of this letter). Furthermore, nothing herein shall serve
as a general waiver or be construed as amending or revising in any way the
definition of "Designated Area" in the Agreement or any other provision of the
Agreement.

     By signing below, you agree that nothing in this letter terminates or
revokes any prior waiver of the Agreement previously granted by you.

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April 23,2002
Page 3

     Please sign this letter in the space below, fax a signed copy to Joe
Dannenmaier of Thompson & Knight L.L.P. (fax 214/ 969-1751) and return the
signed original to Pure Resources, Inc. c/o Joe Dannenmaier, Thompson & Knight
L.L.P., 1700 Pacific Avenue, Suite 3300, Dallas, Texas 75201.

                                   Very truly yours,

                                   PURE RESOURCES, INC.


                                   By:  /s/ DAN P. COLWELL
                                       -----------------------------------------
                                        Dan P. Colwell
                                        Vice President
                                        Acquisitions, Divestures & Land

Accepted and agreed to:

UNION OIL COMPANY OF CALIFORNIA


By:  /s/ DOUGLAS M. MILLER
    ------------------------------
    Name:  Douglas M. Miller
          ------------------------
    Title:  Vice President
           -------------------------